DelaunayTriangulation.jl

Overview

• Delaunay triangulations
• Constrained Delaunay triangulations
• Mesh refinement
• Voronoi tessellations
• Interpolation and solving PDEs

Delaunay Triangulations

• A triangulation of a set of points \(\mathcal P\) in the plane such that no triangle’s circumcircle contains any other point in the set.

Delaunay Triangulations

• A triangulation of a set of points \(\mathcal P\) in the plane such that no triangle’s circumcircle contains any other point in the set.
• Maximizes the minimum angle over all other triangulations of \(\mathcal P\).

Delaunay Triangulations

• A triangulation of a set of points \(\mathcal P\) in the plane such that no triangle’s circumcircle contains any other point in the set.
• Maximizes the minimum angle over all other triangulations of \(\mathcal P\).
• Minimises the worst-case pointwise piecewise linear interpolation error for curvature-bounded functions.

Bowyer-Watson Algorithm

• Triangulations are computed incrementally.

Bowyer-Watson Algorithm

• Triangulations are computed incrementally.

Bowyer-Watson Algorithm

• Triangulations are computed incrementally.

Bowyer-Watson Algorithm

• Triangulations are computed incrementally.

Ghost Vertices

• How to handle points outside of the triangulation?
• Ghost vertices.

Ghost Vertices

• How to handle points outside of the triangulation?
• Ghost vertices.

Ghost Vertices

• How to handle points outside of the triangulation?
• Ghost vertices.

Ghost Vertices

• How to handle points outside of the triangulation?
• Ghost vertices.

Using DelaunayTriangulation.jl

using DelaunayTriangulation
points = [randn(2) for _ in 1:20]
tri = triangulate(points)

Using DelaunayTriangulation.jl

using DelaunayTriangulation
points = [randn(2) for _ in 1:20]
tri = triangulate(points)
add_point!(tri, 0.3, 0.5)

Using DelaunayTriangulation.jl

using DelaunayTriangulation
points = [randn(2) for _ in 1:20]
tri = triangulate(points)
add_point!(tri, 0.3, 0.5)
using CairoMakie
triplot(tri)

Point Location

• At each stage, we need to find the triangle \(T\) containing the inserted point \(q\).
• Jump and march. Accelerate by sampling \(\mathcal O(n^{1/3})\) vertices to start from.

Using DelaunayTriangulation.jl

using DelaunayTriangulation
points = rand(2, 500)
tri = triangulate(points)
find_triangle(tri, (0.2, 0.3))

(203, 258, 30)

Using DelaunayTriangulation.jl

using DelaunayTriangulation
points = rand(2, 500)
tri = triangulate(points)
find_triangle(tri, (0.2, 0.3))
find_triangle(tri, (1.2, 1.5))

(219, 198, -1)

Using DelaunayTriangulation.jl

using DelaunayTriangulation
points = rand(2, 500)
tri = triangulate(points)
find_triangle(tri, (0.2, 0.3))
find_triangle(tri, (1.2, 1.5))
find_triangle(tri, (0.5, 0.5); k=7)

(330, 490, 248)

Constrained Triangulations

• Triangulation of points, segments, and boundaries.
• Useful for enforcing constraints and setting up domains for PDEs.

Constrained Delaunay Property

• A triangle is constrained Delaunay if its circumcircle contains no points in its interior visible from the triangle.
• The edge \(e\) and triangle \(T\) are both constrained Delaunay.

Inserting Segments

• Insert segments one at a time.
• Observation: Segments split the triangulation into two localised parts on either side.

Inserting Segments

• Insert segments one at a time.
• Observation: Segments split the triangulation into two localised parts on either side.

Inserting Segments

• Insert segments one at a time.
• Observation: Segments split the triangulation into two localised parts on either side.

Inserting Segments

• Insert segments one at a time.
• Observation: Segments split the triangulation into two localised parts on either side.

Inserting Segments

• Insert segments one at a time.
• Observation: Segments split the triangulation into two localised parts on either side.

Inserting Segments

• Insert segments one at a time.
• Observation: Segments split the triangulation into two localised parts on either side.

Using DelaunayTriangulation.jl

using DelaunayTriangulation, CairoMakie
points = [(0.0, 0.0), (0.0, 1.0), (0.0, 2.5), (2.0, 0.0),
    (6.0, 0.0), (8.0, 0.0), (8.0, 0.5), (7.5, 1.0),
    (4.0, 1.0), (4.0, 2.5), (8.0, 2.5)]
tri = triangulate(points; segments=Set([(2, 7)]))

triplot(tri)

Using DelaunayTriangulation.jl

using DelaunayTriangulation, CairoMakie
points = [(0.0, 0.0), (0.0, 1.0), (0.0, 2.5), (2.0, 0.0),
    (6.0, 0.0), (8.0, 0.0), (8.0, 0.5), (7.5, 1.0),
    (4.0, 1.0), (4.0, 2.5), (8.0, 2.5)]
tri = triangulate(points; segments=Set([(2, 7)]))
add_segment!(tri, 2, 11)
triplot(tri)

Boundaries

• Boundaries are just sequences of segments.
• Exterior vertices are deleted, identified recursively.

Boundaries

• Boundaries are just sequences of segments.
• Exterior vertices are deleted, identified recursively.

Boundaries

• Boundaries are just sequences of segments.
• Exterior vertices are deleted, identified recursively.

Boundaries

• Boundaries are just sequences of segments.
• Exterior vertices are deleted, identified recursively.

Using DelaunayTriangulation.jl

using DelaunayTriangulation, CairoMakie
boundary = [(0.0, 0.0), (0.1, -0.1), (0.2, -0.2), (0.2, 0.0),
    (0.8, 0.0), (0.8, -0.2), (0.9, -0.1), (1.0, 0.0),
    (0.9, 0.2), (0.7, 0.3), (0.6, 0.2), (0.5, 0.1),
    (0.4, 0.2), (0.3, 0.3), (0.1, 0.2), (0.0, 0.0)]
points = [(rand(), 0.5rand() - 0.2) for _ in 1:1000]
boundary_nodes, points = convert_boundary_points_to_indices(
    boundary; existing_points=points)
tri = triangulate(points; boundary_nodes)
triplot(tri)

Identifying Boundaries

• To support boundary conditions, we need a way to assign IDs to boundaries.
• Define boundaries as sequences of individual sections, with each section having its own ghost vertex.
• Leveraged by FiniteVolumeMethod.jl.

Using DelaunayTriangulation.jl

using DelaunayTriangulation, CairoMakie
section_1 = [(0.0, 0.0), (0.1, -0.1)]
section_2 = [(0.1, -0.1), (0.2, -0.2), (0.2, 0.0),
    (0.8, 0.0), (0.8, -0.2)]
section_3 = [(0.8, -0.2), (0.9, -0.1), (1.0, 0.0)]
section_4 = [(1.0, 0.0), (0.9, 0.2), (0.7, 0.3), (0.6, 0.2), (0.5, 0.1),
    (0.4, 0.2), (0.3, 0.3), (0.1, 0.2), (0.0, 0.0)]
boundary = [section_1, section_2, section_3, section_4]
points = [(rand(), 0.5rand() - 0.2) for _ in 1:1000]
boundary_nodes, points = convert_boundary_points_to_indices(
    boundary; existing_points=points)
tri = triangulate(points; boundary_nodes)

Using DelaunayTriangulation.jl

• You can access features of the boundary easily.

get_boundary_nodes(tri)

4-element Vector{Vector{Int64}}:
 [1001, 1002]
 [1002, 1003, 1004, 1005, 1006]
 [1006, 1007, 1008]
 [1008, 1009, 1010, 1011, 1012, 1013, 1014, 1015, 1001]

Using DelaunayTriangulation.jl

• You can access features of the boundary easily.

get_boundary_nodes(tri)

4-element Vector{Vector{Int64}}:
 [1001, 1002]
 [1002, 1003, 1004, 1005, 1006]
 [1006, 1007, 1008]
 [1008, 1009, 1010, 1011, 1012, 1013, 1014, 1015, 1001]

get_adjacent2vertex(tri, -2)

Set{Tuple{Int64, Int64}} with 4 elements:
  (1004, 1003)
  (1006, 1005)
  (1003, 1002)
  (1005, 1004)

Using DelaunayTriangulation.jl

• You can access features of the boundary easily.

get_boundary_nodes(tri)

4-element Vector{Vector{Int64}}:
 [1001, 1002]
 [1002, 1003, 1004, 1005, 1006]
 [1006, 1007, 1008]
 [1008, 1009, 1010, 1011, 1012, 1013, 1014, 1015, 1001]

get_adjacent2vertex(tri, -2)

Set{Tuple{Int64, Int64}} with 4 elements:
  (1004, 1003)
  (1006, 1005)
  (1003, 1002)
  (1005, 1004)

get_neighbours(tri, -2)

Set{Int64} with 5 elements:
  1005
  1002
  1006
  1003
  1004

Using DelaunayTriangulation.jl

• Disjoint domains and holes follow the same representation.

using DelaunayTriangulation
using CairoMakie
θ = LinRange(0, 2pi, 100);
θ = [θ[1:99]; θ[1]];
θr = reverse(θ)
otrx, otry = 10cos.(θ), 10sin.(θ)
innx, inny = 5cos.(θr), 5sin.(θr)
inn2x, inn2y = 2.5cos.(θ), 2.5sin.(θ)
x = [[otrx], [innx], [inn2x]]
y = [[otry], [inny], [inn2y]]
points = [(20rand() - 10, 20rand() - 10)
          for _ in 1:1000]
boundary_nodes, points =
    convert_boundary_points_to_indices(
        x, y; existing_points=points)
tri = triangulate(points;
    boundary_nodes)
triplot(tri)

Mesh Refinement

• Solutions using meshes are highly dependent on the quality of the triangles.
• Poor quality triangles can cause interpolation errors, gradient errors, and ill-conditioning.

Assessing Element Quality

• Assess triangles using the radius-edge ratio \(\rho = R/\ell_{\min}\).

• \(1/\sqrt 3 \leq \rho <\infty\), with \(1/\sqrt 3\) attained for equilateral triangles.

• Related to the smallest angle by

\[ \rho = \frac{1}{2\sin \theta_{\min}}. \]

Thus, controlling \(\rho\) controls the smallest angle.

Mesh Encroachment

• Mesh refinement inserts points into the triangulation.
• One issue with insertion is encroachment.

Ruppert’s Algorithm

function refine(tri)
    split_all_encroached_segments!(tri)
    while has_bad_triangles(tri)
        T = get_bad_triangle(tri)
        insert_circumcenter!(tri, T)
        if circumcenter_encroaches(tri)
            undo_insertion!(tri, T)
            split_all_encroached_segments!(tri)
        else
            for V in new_triangles(tri)
                ρ, A = radius_edge_ratio(V), area(V)
                (ρ > ρₘₐₓ || A > Aₘₐₓ) && mark_bad!(tri, V)
            end
        end
    end
end

• Typically converges for \(\theta_{\min} < 33.9^\circ\).
• Extra care needed for small angles.

Ruppert’s Algorithm

Using DelaunayTriangulation.jl

using DelaunayTriangulation, CairoMakie
outer = [(0.0, 0.0), (1.0, 0.0), (1.0, 1.0), (0.0, 1.0), (0.0, 0.0)]
inner = [(0.25, 0.25), (0.25, 0.75), (0.75, 0.75), (0.75, 0.25), (0.25, 0.25)]
boundary = [[outer], [inner]]
boundary_nodes, points = convert_boundary_points_to_indices(boundary)
segments = Set([(1, 5), (7, 3)])
tri = triangulate(points; boundary_nodes, segments)
refine!(tri; max_area=0.01)
triplot(tri)

Curve-Bounded Domains

• Generalise triangulations to allow for parametric curves.
• Six types of parametric curves are supported, but the interface can be extended easily to new curves.

Boundary Enrichment

• Equally spaced discretisations can lead to excess triangles. Use curvature.
• Discretise until all edges are constrained Delaunay and the total variation \(\int_C |\kappa(s)|\,\mathrm ds < \pi/2\) over each subcurve.

Boundary Enrichment

• Equally spaced discretisations can lead to excess triangles. Use curvature.
• Discretise until all edges are constrained Delaunay and the total variation \(\int_C |\kappa(s)|\,\mathrm ds < \pi/2\) over each subcurve.

Boundary Enrichment

• Equally spaced discretisations can lead to excess triangles. Use curvature.
• Discretise until all edges are constrained Delaunay and the total variation \(\int_C |\kappa(s)|\,\mathrm ds < \pi/2\) over each subcurve.

Curve-Bounded Refinement

• Works just as in the standard case, except encroached edges are split at the equivariation point instead of their midpoint.

Comparison with Discretisation

Using DelaunayTriangulation.jl

Random.seed!(123)
using DelaunayTriangulation, CairoMakie
r₁, r₂, r₃ = 10.0, 5.0, 2.5
c₁ = CircularArc((r₁, 0.0), (r₁, 0.0), (0.0, 0.0))
c₂ = CircularArc((r₂, 0.0), (r₂, 0.0), (0.0, 0.0), positive=false)
c₃ = CircularArc((r₃, 0.0), (r₃, 0.0), (0.0, 0.0))
tri = triangulate(NTuple{2,Float64}[]; boundary_nodes=[[[c₁]], [[c₂]], [[c₃]]])
refine!(tri; max_area=0.001get_area(tri))
triplot(tri)

Heterogenous Constraints

• Use custom_constraint to apply different constraints in each region.

Random.seed!(123)
outer_area = π * (r₁^2 - r₂^2)
inner_area = π * r₃^2
function area_constraint(tri, T)
    u, v, w = triangle_vertices(T)
    p, q, r = get_point(tri, u, v, w)
    A = DelaunayTriangulation.triangle_area(p, q, r)
    c = (p .+ q .+ r) ./ 3
    poly = find_polygon(tri, c) # 1: outer-most; 3: inner-most; 2: in-between
    return poly == 1 ? (A ≥ 0.005inner_area) : (A ≥ 0.001outer_area) # 2 is impossible
end
tri = triangulate(NTuple{2,Float64}[]; boundary_nodes=[[[c₁]], [[c₂]], [[c₃]]])
refine!(tri; custom_constraint=area_constraint)
triplot(tri)

Heterogenous Constraints

Voronoi Tessellations

• Dual graph to the Delaunay triangulation.
• Each point in a Voronoi cell is closer to its associated generator than to any other generator.

Using DelaunayTriangulation.jl

using DelaunayTriangulation, CairoMakie
points = randn(2, 25)
tri = triangulate(points)
vorn = voronoi(tri)
voronoiplot(vorn)

Clipped Voronoi Tessellations

• Support for clipping to convex polygons.

vorn = voronoi(tri; clip=true) # clips to the convex hull
voronoiplot(vorn)

Clipped Voronoi Tessellations

• Support for clipping to convex polygons.

clip_points = [(-2.0, -1.0), (2.0, -1.0), (2.0, 1.0), (-2.0, 1.0)]
clip_vertices = [1, 2, 3, 4, 1]
vorn = voronoi(tri; clip=true,
    clip_polygon=(clip_points, clip_vertices)) # clips to the square
voronoiplot(vorn)

Clipped Voronoi Tessellations

• Support for clipping to convex polygons.

ellip = EllipticalArc((2.0, 0.0), (2.0, 0.0), (0.0, 0.0), 2.0, 0.8, 0.0)
t = LinRange(0, 1, 100)
clip_points = ellip.(t)[1:end-1]
clip_vertices = [1:length(clip_points); 1]
vorn = voronoi(tri; clip=true,
    clip_polygon=(clip_points, clip_vertices)) # clips to the ellipse
voronoiplot(vorn)

Centroidal Voronoi Tessellations

• Support for centroidal Voronoi tessellations.

points = rand(2, 500)
vorn = voronoi(triangulate(points); clip=true, smooth=true)
voronoiplot(vorn)

FiniteVolumeMethod.jl

• FiniteVolumeMethod.jl leverages DelaunayTriangulation.jl for meshing, solving PDEs of the form \[ \frac{\partial u(\boldsymbol x, t)}{\partial t} + \boldsymbol \nabla \cdot \boldsymbol q(\boldsymbol x, t, u) = S(\boldsymbol x, t, u) \]
• Hooks into the SciML interface to support a wide range of solvers.

FiniteVolumeMethod.jl

NaturalNeighbours.jl

• NaturalNeighbours.jl uses DelaunayTriangulation.jl for performing natural neighbour interpolation.

Natural Neighbours

• Use interpolants of the form \(f(\boldsymbol x_0) = \sum_{i \in N_0} \lambda_i f(\boldsymbol x_i)\).
• Sibson coordinates: \(\lambda_i = A(\boldsymbol x_i)/A(\boldsymbol x)\).

Conclusion

• Delaunay triangulations are a powerful tool for meshing and interpolation, among other applications.
• DelaunayTriangulation.jl is a feature-rich package for working with Delaunay triangulations and Voronoi tessellations.
• Documentation contains many other examples, details, and applications such as cell biology.
• Experimental support for spherical Delaunay triangulations and Voronoi tessellations.
• Also supports weighted triangulations and power diagrams.